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PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
Oxygen vacancies in strontium titanate: A DFT+DMFT study
Jaime Souto-Casares,* Nicola A. Spaldin ,† and Claude Ederer ‡
Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
(Received 17 December 2020; accepted 16 March 2021; published 9 April 2021)
We address the long-standing question of the nature of oxygen vacancies in strontium titanate, using a combi-
nation of density functional theory and dynamical mean-field theory (DFT+DMFT) to investigate in particular
the effect of vacancy-site correlations on the electronic properties. Our approach uses a minimal low-energy
electronic subspace including the Ti-t2g orbitals plus an additional vacancy-centered Wannier function, and it
provides an intuitive and physically transparent framework to study the effect of the local electron-electron
interactions on the excess charge introduced by the oxygen vacancies. We estimate the strength of the screened
interaction parameters using the constrained random phase approximation, and we find a sizable Hubbard U
parameter for the vacancy orbital. Our main finding, which reconciles previous experimental and computational
results, is that the ground state is either a state with double occupation of the localized defect state or a state with
a singly occupied vacancy and one electron transferred to the conduction band. The balance between these two
competing states is determined by the strength of the interaction both on the vacancy and the Ti sites, and on the
Ti-Ti distance across the vacancy. Finally, we contrast the case of vacancy doping in SrTiO3 with doping via La
substitution, and we show that the latter is well described by a simple rigid-band picture.
DOI: 10.1103/PhysRevResearch.3.023027
I. INTRODUCTION question is the nature of the introduced charge, with different
experimental measurements leading to apparently contradic-
Strontium titanate, SrTiO3 , is a perovskite-structure oxide
tory conclusions. On the one hand, it is known that even
with the ideal cubic Pm3̄m structure at room temperature,
very low concentrations of oxygen vacancies cause metal-
and a band-insulating electronic structure due to the formal
licity, with transport measurements indicating an increase in
3d 0 configuration of the Ti4+ cations. In spite of its apparent
carrier density with increasing oxygen vacancy concentration
simplicity, SrTiO3 shows a wealth of interesting and some-
[23–25]. On the other hand, there are multiple reports of
times technologically relevant properties, such as tunability
optical absorption signals within the band gap [26–29], such
of its high dielectric constant [1,2], quantum paraelectricity
as, for example, a red luminescence at 2.0 eV, attributed to
[3,4], and even superconductivity [5]. Although these prop-
localized electrons forming Ti3+ polarons trapped at isolated
erties have been known for many years, there remain many
oxygen vacancies [30].
open questions [6]. For example, superconductivity occurs at
Computational studies aiming to clarify the physics of the
unusually low doping levels [7], indicating an exotic mech-
OV state also show a range of conflicting scenarios, largely
anism possibly related to its quantum paraelectricity [8]. At
due to their different treatments of exchange and correlation
the same time, reports of a two-dimensional electron gas [9]
effects. (For a detailed summary, see Ref. [31].) Standard
and emergent magnetism [10] at surfaces and interfaces have
density functional calculations using the local density ap-
rekindled interest in SrTiO3 thin films for oxide electronics.
proximation (LDA) or generalized gradient approximation
All of these phenomena require the existence of electronic
(GGA) predict a delocalized defect state at the bottom of the
charge carriers, which are usually introduced through oxygen
Ti-t2g conduction band, consistent with the observed metallic-
vacancies (OV ) [11], substitution of Sr 2+ by a trivalent ion
ity [32–34]. Use of B3PW hybrid functionals or the LDA+ U
such as La3+ , or of Ti4+ by a pentavalent ion such as Nb5+ .
method, however, lead to a doubly occupied in-gap state,
Interestingly, the resulting properties can be quite sensitive
0.77 eV (B3PW) or 0.11 eV (LDA+ U , with U = 5 eV)
to the specific type of doping [12–22]. Regarding oxygen-
below the minimum of the conduction band, but do not cap-
vacancy doping in SrTiO3 , perhaps the most pressing open
ture the reported metallic conduction (although this can be
reconciled by considering the formation of polarons) [35].
Using spin-polarized GGA+ U calculations with U = 5.0 eV,
* the two electrons released by the missing oxygen are found
jaime.soutocasares@mat.ethz.ch
†
nicola.spaldin@mat.ethz.ch to distribute between a localized magnetic in-gap state and
‡
claude.ederer@mat.ethz.ch a delocalized state in the conduction band, consistent with
both sets of reported experimental behaviors [31]. Finally,
Published by the American Physical Society under the terms of the we mention a recent DFT + U + V study, including self-
Creative Commons Attribution 4.0 International license. Further consistent on-site and intersite electronic interactions, which
distribution of this work must maintain attribution to the author(s) gives a good description of both stoichiometric and oxygen-
and the published article’s title, journal citation, and DOI. deficient STO, with the details of the band structure and the
2643-1564/2021/3(2)/023027(11) 023027-1 Published by the American Physical Society
SOUTO-CASARES, SPALDIN, AND EDERER PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
vacancy formation energies agreeing well with experiments us to explore in detail the effect of the electronic correlations
[36]. The authors also present a systematic study of the ef- on the vacancy site, and it is the main novelty of the method-
fects of various parameters such as structure and cell size, ology presented here.
the used exchange-correlation functional, and the treatment To obtain accurate geometries and initial band structures,
of structural relaxations and spin polarization on the resulting we perform standard DFT calculations using the projector-
properties. augmented-wave (PAW) method, as implemented in the
These previous studies clearly indicate that electron in- Vienna Ab-initio Simulation Package (VASP) [47,48], version
teraction effects are important in describing the behavior 5.4.1, together with the LDA exchange-correlation functional
of oxygen vacancies in SrTiO3 [37], in spite of the band- [49]. The valence configurations of the PAW potentials used
insulating nature of the stoichiometric host material. However, are Sr(3s2 3p6 4s2 ), Ti(3s2 3p6 4s1 3d 3 ), and O(2s2 2p4 ). To al-
while such DFT+ U (+V ) calculations are able to appro- low the system to accommodate the defect with a reasonable
priately treat strong interactions between electrons in the OV concentration, we use a 20-atom unit cell, 19-atom for
transition-metal d states (and in principle the oxygen p states), the OV -defective system SrTiO2.75 , corresponding to an oxy-
this is usually achieved by introducing an artificial symmetry gen vacancy concentration of 8.3% [50]. Our calculations for
breaking resulting in long-range magnetic order. Further- Lax Sr 1−x TiO3 (Sec. III F) are performed using larger 40-atom
more, current implementations of the DFT+ U formalism are supercells. For calculations of geometries and band structures
typically based on projections on site-centered atomic or- (Secs. III A and III B), well-converged results are obtained
bitals, and so they leave electrons at the vacancy site by sampling the Brillouin zone with an 8 × 8 × 8 -centered
uncorrelated. The importance of explicitly considering inter- k-mesh and using a plane-wave energy cutoff of 800 eV.
actions on the vacancy site was shown in model calculations For the more demanding cRPA calculations (Sec. III D), the
using a minimal three-orbital model representing the vacancy, k-space sampling is done with a 4 × 4 × 4 mesh, and an en-
coupled to a bath representing the t2g bulk bands [38], but a ergy cutoff of 500 eV is used. In Sec. III E (Ti-OV -Ti distance
full first-principles description is still lacking. dependence), these parameters are 8 × 8 × 8 and 750 eV, and
Here, we study the electronic properties of oxygen- in Sec. III F (comparison with La doping), these are 4 × 4 × 4
deficient SrTiO3 using a combination of density functional and 700 eV. Where lattice relaxation is employed, structural
theory (DFT) and dynamical mean-field theory (DMFT). The degrees of freedom are relaxed until forces fall below a 10−4
latter accounts for all dynamic correlation effects between eV/Å threshold, with the symmetry constrained to disable the
electrons on the same “site” and thus introduces genuine rotations of the oxygen octahedra that occur in SrTiO3 below
many-body effects in the electronic structure obtained from ∼100 K. All calculations are performed with spin polarization
DFT, and it has already been applied recently to study oxygen excluded.
vacancy complexes at the SrTiO3 surface [39]. Similar to our The low-energy correlated subspace for the DMFT calcula-
previous work on oxygen-deficient LaTiO3 [40], we focus on tions is then constructed using a basis of maximally localized
the most important bands at the bottom of the conduction Wannier functions (MLWF) [41,51], employing the WAN-
band, which can be expressed in a basis of maximally local- NIER90 code [42]. We use the TRIQS/DFTTOOLS package
ized Wannier functions (MLWFs) [41,42] with predominant [52–54] to implement the DMFT calculations, averaging over
Ti-t2g character plus one additional Wannier function located both spin channels to enforce a paramagnetic solution. An ef-
at the vacancy site. This results in a physically intuitive frame- fective impurity problem is solved for each inequivalent Ti site
work to study the distribution of charge between the vacancy plus the vacancy site using the TRIQS/CTHYB solver [55],
site and the Ti-t2g conduction bands. while the different impurity problems are coupled through
We estimate the strength of the screened electron-electron the DMFT self-consistency. The OV site is treated at the
interaction, both for the Ti-t2g orbitals and for the vacancy same level as the Ti sites, as introduced in Ref. [40] (more
level, using the constrained random phase approximation details about this implementation are given in Sec. III C).
(cRPA) [43–46]. We obtain a sizable U parameter for the The DFT+DMFT calculations are performed without full
vacancy level, confirming that the corresponding correlation charge self-consistency. The local interaction is modeled
effects should not be ignored. We then find that the occupation using the Hubbard-Kanamori parametrization with spin-flip
of the vacancy site is indeed controlled by the strength of and pair-hopping terms included [56]. Within the Hubbard-
the electron-electron repulsion both on the OV as well as on Kanamori parametrization, the strength of the electron-
the Ti sites, balancing the system between a doubly occupied electron interaction is described by the intraorbital Hubbard
localized vacancy state and a singly occupied vacancy with the parameter U and the Hund coupling parameter J. Note that
other electron doping the conduction band. The latter case can these parameters are different from the average interaction
also be viewed as a site-selective Mott insulator, in which the parameters U and J typically used in DFT+ U calculations.
singly occupied in-gap state corresponds to the lower Hubbard The double-counting correction is computed within the fully
band on the vacancy site. localized limit according to Held [57], and all calculations are
performed at room temperature, β = (kB T )−1 = 40 eV−1 . We
use a fixed value of J = 0.64 eV on the Ti sites, whereas the
II. COMPUTATIONAL METHOD
values for the Hubbard U , both on the vacancy and the Ti sites,
The key point in our analysis is the explicit inclusion of an are varied to analyze the effect on the electronic properties.
electronic orbital corresponding to the oxygen vacancy point Full frequency spectral functions, A(ω), are obtained from
defect into the low-energy Hamiltonian constructed from the the local Green’s functions in imaginary time, G(τ ), using
DFT results, together with the usual Ti-t2g bands. This allows the Maximum Entropy algorithm [58]. The spectral weight
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OXYGEN VACANCIES IN STRONTIUM TITANATE: A … PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
we keep the lattice constants fixed to our calculated values
for stoichiometric SrTiO3 [60]. The resulting structure and
Ti-Ti distances are shown in Fig. 1(c). The removal of one of
the oxygen atoms lowers the symmetry of the formerly cubic
crystal to tetragonal P4/mmm, in which the unique axis corre-
sponds to the Ti-OV -Ti chain [y axis in Fig. 1(b)], and divides
the Ti sites into two inequivalent types: one next to the va-
cancy, with five Ti-O bonds, and one farther from the vacancy,
with a complete oxygen coordination octahedron. (Note that
we do not include the low-temperature antiferrodistortive ro-
tations of the oxygen octahedra; our preliminary tests suggest
that their effect on the vacancy state is rather weak.) The Ti-Ti
distances in the plane perpendicular to the tetragonal axis are
slightly modified, with the vacancy pushing the two closest
Ti apart, increasing their distance by 2.5% with respect to the
distance in stoichiometric SrTiO3 . The next Ti-O-Ti distance
on the same axis contracts by the same amount, due to the
volume constraint. On the other inequivalent Ti-O-Ti chain
along the y axis there is also a 1.6% contraction (expansion)
of the Ti-O-Ti distance at the same (different) y position as the
closest Ti-OV -Ti.
FIG. 1. Supercells of the calculated structures for (a) SrTiO3
B. DFT band structures
(20-atom unit cell, Pm3̄m symmetry) and (b) SrTiO3−δ (19-atom
unit cell, P4/mmm symmetry). (c) Geometry of the relaxed structure The calculated band structures for stoichiometric SrTiO3
containing an oxygen vacancy (indicated with the orange circle) and defective SrTiO3−δ are shown and compared in Fig. 2.
with the Ti-Ti distances given in Å. For comparison, the calculated The low-energy region around the gap is composed of a va-
Ti-Ti distance for stoichiometric SrTiO3 is 3.862 Å. Note that the lence band of mostly O-p character and a conduction band
orientation of part (c) is rotated relative to that of parts (a) and (b). whose bottom part has mainly Ti-t2g contributions, with some
weight coming from the O-p orbitals, and minimal overlap
with the Ti-eg and Sr-s bands at ∼3 eV above the gap. Va-
around the Fermi energy, Ā(0), is calculated from the impurity lence and conduction bands are separated by a gap of 1.8 eV,
Green’s function as Ā(0) = −β/π G(β/2). The quasiparticle strongly underestimating the experimental value of 3.25 eV
weight, Z, is calculated for each site as Z = [1 − (iω0 )]−1 , [61]. The bands between approximately 0 and 3 eV can be
where (iω0 ) is its self-energy at ω0 , the smallest calculated expressed in terms of MLWFs centered on the Ti atoms and
Matsubara frequency. showing a strong t2g orbital character with additional admix-
We calculate the screened Coulomb interaction within the tures of O-p on the surrounding ligands; see Fig. 2(d).
low-energy correlated subspace using the constrained random The removal of one oxygen atom changes mainly the lower
phase approximation (cRPA) [43–46]. In the cRPA method, part of the conduction-band region of the band structure, with
a partial polarization function, Pr , is calculated by exclud- the appearance of an additional band that crosses the Fermi
ing all possible electronic transitions taking place within energy, EF , making the system metallic, and accompanied by
the correlated subspace. The bare Coulomb interaction, v, a lifting of degeneracies at the special points of the Brillouin
is then renormalized through screening by the higher-energy zone. This band could accommodate, in principle, the two
degrees of freedom through this Pr , yielding the frequency- electrons released by the vacancy. However, its overlap with
dependent partially screened Coulomb interaction Wr (ω) = the bottom of the conduction-band edge (Fig. 2) results in a
[1 − vPr (ω)]−1 v. The local interaction parameters of the partial transfer of charge into the Ti-t2g bands. Therefore, a
Kanamori Hamiltonian, U and J, are then obtained from the complete description of the low-energy behavior of the system
static limit by calculating matrix elements of Wr (ω = 0), with must include both the Ti-t2g bands and this OV -induced band,
the MLWFs used as local basis orbitals within the DMFT and we proceed by incorporating it into the minimal basis of
calculation, and taking appropriate averages over orbitals. Ti-t2g-like MLWFs in the DMFT treatment. We thus construct
12 + 1 MLWFs, using initial t2g projections on the Ti sites
III. RESULTS AND DISCUSSION plus one s-like projection centered around the vacancy site.
The relationship of this new band to the OV is clearly
A. Geometry optimization seen from the MLWF centered on the position of the missing
Our calculated optimized LDA lattice constant for SrTiO3 oxygen: its real-space representation shows an approximately
constrained to cubic Pm √3̄m symmetry
√ is 3.862 Å. [For a spherical orbital centered around the vacancy with tails reach-
picture of the 20-atom ( 2, 2, 2) unit, see Fig. 1(a).] This ing to the neighboring ions [Fig. 2(e)]. Moreover, if one
value agrees well with literature LDA calculations [34], and it extracts a single MLWF centered on the vacancy site, the
is not too far from the experimental value of 3.900 Å [59]. In resulting Wannier band closely follows the new OV -induced
subsequent relaxations after removing one oxygen [Fig. 1(b)], Bloch band [orange thick line in Fig. 2(b)]. As one can see
023027-3
SOUTO-CASARES, SPALDIN, AND EDERER PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
repulsion. This procedure allows us to independently vary the
strength of the local interaction on the different types of sites,
including the vacancy, and hence to determine the evolution of
the system when U (OV ) changes independently of U (Ti). We
note that for other early transition-metal perovskites, such as,
e.g., LaTiO3 , values of U (Ti) between 4 and 5 eV have often
led to good agreement with experimental observations when
using a minimal t2g orbital subspace [62,63]. We therefore
choose the range from 2 to 8 eV. On the other hand, given the
lack of chemical intuition for choosing U (OV ), we have con-
sidered U (OV ) = 0 and U (OV ) =U (Ti) as reasonable limits.
Figure 3 shows our calculated DMFT site occupations,
along with the site-resolved quasiparticle weight Z, and the
corresponding spectral functions for different choices of U on
the Ti and the OV sites. The top row [Fig. 3(a)] corresponds
to U (OV ) = 0, which is treating the electrons on the vacancy
site as “uncorrelated.” In this case, the effect of increasing
U (Ti) is to increase the OV occupation from 6 eV, a metal-insulator transition
(MIT) is observed in which the Ti bands are totally depleted
and the system becomes a band insulator. This MIT is related
to a shift of the OV (Ti) spectral weight to lower (higher)
energies until the corresponding overlap vanishes [see the
spectral function for U (Ti) = 8 eV]. The quasiparticle weight
Z is close to 1 for all sites, independent of U (Ti), indicating
that the electrons remain uncorrelated, consistent with the
transition from an uncorrelated metal to a band insulator [64].
Our DFT+DMFT result for higher U (Ti) and U (OV ) = 0
FIG. 2. Calculated DFT band structures for (a) stoichiometric
is thus equivalent to the results obtained in previous studies
SrTiO3 and (b) SrTiO2.75 . DFT and MLWF bands are shown as gray within DFT+ U [32], as discussed in Sec. I.
and blue solid lines, respectively; the vacancy band in (b) is high- The remaining rows in Figs. 3(b)–3(e) demonstrate the
lighted in orange (see the main text). (c) MLWF-projected density effect of introducing and then increasing a local Coulomb re-
of states (DOS) for the t2g-like and OV MLWFs. Part (d) shows the pulsion on the vacancy site, with U (OV ) U (Ti). By follow-
real-space representation of a dxz -type MLWF in the stoichiometric ing the evolution of the OV site occupation as U (OV )/U (Ti)
SrTiO3 system, while (e) corresponds to the OV MLWF. grows, we can see that the main effect of U (OV ) is to first
weaken the effect of U (Ti) toward establishing a doubly occu-
pied vacancy site [see the cases with 0 < U (OV ) U (Ti)/2 in
from the MLWF-projected density of states [Fig. 2(c)], the Figs. 3(b) and 3(c)], and then, for U (OV ) >U (Ti)/2, to drive
OV -centered MLWF accounts for most of the weight of this the system instead toward a state with a half-filled OV and one
additional band. The occupations of the Wannier states, as electron doped into the Ti bands; see Figs. 3(d) and 3(e). In
calculated in DFT, are 0.21, 0.28, and 1.02 for the two in- this limit, the Ti sites obtain an average filling of about 0.25,
equivalent Ti sites, next to and farther from the OV , and the with a slightly higher occupation of the Ti farther away from
OV site itself, respectively, in units of the electron charge. the vacancy.
Increasing U (OV ) also strongly reduces the quasiparticle
weight on the vacancy site, indicating strong local electronic
C. DMFT results correlations. Simultaneously, U (OV ) triggers a reduction of
Next, we perform DMFT calculations to investigate how an Z on the Ti sites, in particular on the site farther away from
explicit local Hubbard-like interaction between the electrons the vacancy. One can also observe a pronounced effect on
affects the spectral properties and the charge distribution be- the site-resolved spectral functions. In particular, increasing
tween the vacancy state and the Ti-t2g bands. The input for U (OV ) results in a clear gap opening in the spectral function
the DMFT calculations is constructed from the tight-binding- on the vacancy site once it reaches half-filling. This indicates
like Hamiltonian expressed in the basis of the MLWFs, plus a site-selective Mott transition, which is a localization of one
the Coulomb matrix that models the electron-electron inter- electron on the vacancy site while the doped Ti bands remain
action. As stated in Sec. II, we simplify the latter for the metallic, albeit with a quasiparticle renormalization on the
t2g states by using the Kanamori form, for which only two farther Ti site of Z ≈ 0.7. Thus, this regime is character-
site-dependent parameters have to be specified: U , the on-site ized by a metallic conduction band doped with one electron
intraorbital Hubbard repulsion, and J, the Hund’s coupling. per vacancy, and a split-off “in-gap” state containing one
For the OV site with only one orbital, there is only one param- electron localized on the vacancy site. In the picture of the
eter, U , describing the corresponding intraorbital Coulomb site-selective Mott insulator, this in-gap state corresponds to
023027-4
OXYGEN VACANCIES IN STRONTIUM TITANATE: A … PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
FIG. 3. DFT+DMFT results for different settings of U(OV )/U(Ti), from 0 (a) to 1 (e). The left column in each row shows the site
occupations for the three types of correlated sites: Ti-next, Ti-farther, and OV (lighter blue, darker blue and orange, respectively). The second
to left column shows the corresponding quasiparticle weights Z. The following three plots in each row show the spectral functions for the three
aforementioned sites for three different U (Ti) values.
the lower Hubbard band of the vacancy site spectral function. with the missing oxygen depends critically on the values
Note that, in this regime of U (OV ) =U (Ti), the filled region of the interaction parameters, U (Ti) and U (OV ). In some
of the OV spectral function overlaps with the O-p bands, cases, values for the interaction parameters can be estimated
which start 2 eV below EF , but are excluded from the DMFT from previous experience or from a comparison of certain
calculation. calculated quantities with corresponding experimental mea-
Our results show that the specific treatment of electronic surements. In the present case, however, there is a lack
correlations on the OV site has a strong influence on whether of consensus among experimental studies (as described in
the two electrons released by the missing oxygen are itiner- Sec. I). Furthermore, due to the use of a nonstandard orbital
ant or localized. Several scenarios that have previously been basis, in particular on the vacancy site, it is not a priori
discussed and reported for SrTiO3−δ can be realized with par- clear what values for U (Ti) and U (OV ) will provide the most
ticular choices of U (Ti) and U (OV ), from the band-insulating realistic description of SrTiO3−δ . Therefore, we now estimate
limit with localization of the vacancy charge in the band the strength of the screened Coulomb repulsion correspond-
gap [65] [high U (Ti), zero U (OV )] to the case of a param- ing to our orbital basis using the constrained random phase
agnetic impurity, in which one electron remains trapped in approximation (cRPA) [43–46].
the gap while the other is delocalized into the Ti d-bands Within cRPA, the electronic degrees of freedom are divided
[U (Ti) =U (OV )] [31]. We note that the lack of charge self- into a “screening subspace” and a “target subspace,” and ex-
consistency in our calculations may diminish the quantitative citations taking place exclusively within the target subspace
nature of these results; however, the qualitative picture should are excluded from the screening (see Sec. II). Generally, the
remain valid. cRPA target subspace should be identical to the correlated
subspace used for the DMFT calculation. However, there
are also cases in which a certain number of “uncorrelated”
D. cRPA calculations bands are included in the DMFT subspace, without consider-
The results discussed in the previous section show that ing a local Hubbard-like interaction for these bands. In such
the localization/delocalization of the two electrons associated cases, it might be appropriate to include these uncorrelated
023027-5
SOUTO-CASARES, SPALDIN, AND EDERER PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
FIG. 4. Averaged values for the partially screened interaction
parameters U (Ti) and U (OV ) obtained within cRPA. Blue dots rep- FIG. 5. Total energy as a function of the Ti-OV -Ti distance (in
resent the values for the Ti sites, while the orange star corresponds units of the Ti-O-Ti distance in stoichiometric SrTiO3 ) obtained
to the OV site. Displayed on the horizontal axis are the different within LDA+U for three different choices of the Hubbard U param-
schemes for choosing the target subspace (see the main text for more eter. The vertical line highlights the distance in cubic stoichiometric
details). SrTiO3
bands within the screening subspace in the cRPA calculation
[44,45]. (t2g/t2g) and (t2g/t2g-OV ) cases is negligible, in spite of the fact
Thus, in order to gain additional insights, and to see how that the average quadratic spread of the Ti-t2g Wannier orbitals
the calculated values depend on the specific subdivision of is reduced from 3.3 to 2.0 Å2 (4.9 Å2 for the OV orbital). How-
electronic degrees of freedom, we consider three different ever, this merely increases the bare (unscreened) interaction
cases. In the first case, we construct only three t2g-like Wan- parameter from 14.1 to 14.9 eV, indicating that the dominant
nier functions for each Ti site. These orbitals then also define factor in determining the final U value is the screening. Thus,
the cRPA target subspace, achieved through a projection of removing the vacancy band from the screening channel has a
the Bloch states onto these Wannier orbitals. All other bands, stronger effect, as can be seen by comparing cases (t2g/t2g-OV )
including that related to the vacancy, act as the screening and (t2g-OV /t2g-OV ), resulting in an increase of U (Ti) from
subspace. Following the notation established in Ref. [44] and 2.65 to 3.47 eV, and demonstrating the sensitivity of U on the
used in several other studies [46,66,67], we denote this case specific screening channel. In addition, a sizable interaction
as (t2g/t2g). Here, the first symbol denotes the orbitals/bands parameter of U (OV ) = 2.12 eV is obtained, which corre-
defining the cRPA target subspace, while the second symbol sponds to approximately 60% of U (Ti). This clearly shows
indicates the whole set of Wannier functions that has been that interaction effects cannot be neglected for the vacancy
constructed. The screened interaction parameters are then orbital, and that, in addition, the presence of the vacancy band
evaluated for the Wannier functions spanning the cRPA target strongly influences the effective interaction parameters on the
subspace. Ti sites. These findings thus further support our approach of
For the second scenario, (t2g-OV /t2g-OV ), a full Wan- including an explicit treatment of correlation effects on the
nier representation of the low-energy conduction bands in vacancy orbital, reinforcing the crucial role played by both
SrTiO3−δ containing three t2g-like Wannier functions per Ti U (Ti) and U (OV ) already presented in Sec. III C.
plus an additional one located on the vacancy site is con- The values of the averaged interaction parameters obtained
structed, as described in the previous section. This whole set for the case (t2g-OV /t2g-OV ), namely U (Ti) ≈ 3.5 eV and
of Wannier functions is then used to define the cRPA target U (OV ) ≈ 2.1 eV, would correspond to DFT+DMFT results
subspace. Thus, in this case the vacancy band is excluded from with a vacancy occupation slightly larger than 1 and a cor-
the screening and included in the target/correlated subspace. responding quasiparticle weight below 0.5, but still with a
The third, intermediate, case is denoted as (t2g/t2g-OV ). metallic spectral function on the vacancy site (Fig. 3). We
Here, the same Wannier functions as in the previous case are note, however, that simply using the static (ω = 0) value
constructed, but only the subset of t2g-like functions are used of the screened interaction in a DFT+DMFT calculation
to define the cRPA target subspace. This essentially means with frequency-independent local interaction might under-
that the effective interaction parameters calculated for the estimate the corresponding interaction effects. Furthermore,
Ti-t2g orbitals also include screening processes involving the recent work has shown that the random phase approximation
vacancy band, similar to the first case, but using exactly the can lead to an “overscreening” of the local interaction [68],
same Wannier representation as in the second scenario. and thus an underestimation of U , in particular for strongly
Figure 4 shows the values of the screened intraorbital correlated systems [69]. The cRPA results should thus rather
interaction parameters U (Ti) and U (OV ), given by the cor- be viewed as providing a rough ballpark, or a lower bound,
responding averaged diagonal elements of the calculated U rather than definite values to be used in realistic DFT+DMFT
tensor for all three cases. The difference in U (Ti) between the calculations [70].
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OXYGEN VACANCIES IN STRONTIUM TITANATE: A … PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
FIG. 6. Evolution of the SrTiO3−δ band structure with respect to the Ti-OV -Ti distance, measured in units of the Ti-O-Ti distance in
stoichiometric SrTiO3 . The central plot corresponds to the Ti-O-Ti distance in stoichiometric SrTiO3 . We see that bringing the two Ti ions
closer together (left side) pushes the OV band down in energy, favoring charge localization, while pulling the Ti atoms apart (right side)
enhances the entanglement of the OV band with the Ti-t2g bands, and redistributes some of its charge onto the Ti ions.
E. Influence of the Ti-OV -Ti distance case. While this is consistent with previous work using LDA
As shown Fig. 1 and discussed in Sec. III A, structural or GGA [71,72], other calculations using hybrid functionals
relaxation using the standard LDA results in an outward re- or LDA+U corrections have instead found a contraction of
laxation of the two Ti atoms adjacent to the vacancy, and thus the Ti-OV -Ti distance, independent of other factors such as
an elongation of the Ti-OV -Ti distance across the vacancy the size of the supercell [73,74]. As mentioned previously
compared to the Ti-O-Ti distance in the ideal stoichiometric (see Sec. I), the latter calculations also find a tendency for
charge localization, whereas LDA/GGA result in some charge
spilling into the Ti bands. Therefore, we next manually tune
the Ti-OV -Ti distance and calculate the resulting changes in
electronic properties.
Figure 5 shows the total energy calculated using LDA and
LDA+ U for 19-atom SrTiO3−δ as a function of the Ti-OV -Ti
distance (in units of the Ti-O-Ti distance in stoichiometric
SrTiO3 ). In these calculations, we only allow for a structural
relaxation of those oxygen atoms that are situated next to the
Ti in the same plane perpendicular to the Ti-OV -Ti direction.
Calculations are performed for three different choices of the
static Hubbard U parameter [75]. The case with U = 0 corre-
sponds to a standard LDA calculation and leads to an increase
of the Ti-OV -Ti distance, consistent with the results presented
in Sec. III A. Increasing U leads to a decrease of the relaxed
Ti-OV -Ti distance, in accordance with the aforementioned
trend in the literature, ultimately shortening this distance with
respect to that in pure SrTiO3 .
To further explore the relationship between the localization
of the defect state and the Ti-OV -Ti distance, we perform
LDA calculations for a SrTiO3−δ structure in which we have
systematically varied the Ti-OV -Ti distance. For each (fixed)
Ti-OV -Ti distance, the oxygen ions are allowed to relax within
the tetragonal symmetry constraint, while all other atoms and
the lattice constant are kept fixed. Figure 6 shows the evo-
lution of the resulting DFT band structure as the Ti-OV -Ti
distance is varied over a range that includes also the var-
ious equilibrium values obtained for different values of U
within DFT+ U . We see that the Ti-OV -Ti distance affects
primarily the position of the vacancy band, which is higher in
energy, and therefore contains less electronic charge, at larger
Ti-OV -Ti distances. We conclude that short Ti-OV -Ti dis-
tances tend to localize the charge around the defect, while
FIG. 7. SrTiO3−δ DMFT occupations for the three different longer distances tend to partially delocalize the charge.
sites as a function of the Ti-OV -Ti distance (in units of the Ti- Following the same procedure as in Sec. III C, we then
O-Ti distance in stoichiometric SrTiO3 ), for three different values perform a series of DMFT calculations for the different
of U (OV )/U (Ti). Lighter blue, darker blue, and orange represent structures corresponding to different Ti-OV -Ti distances. We
Ti-next, Ti-farther, and OV sites, respectively. Electron transfer from use U (Ti) = 4 eV and three different values for U (OV )
the OV to the Ti ions is favored by large Ti-OV -Ti distance and by [0, U (Ti)/2, and U (Ti)]. Our calculated site occupations are
large U (OV ). displayed in Fig. 7. We see that for all values of U (OV ),
023027-7SOUTO-CASARES, SPALDIN, AND EDERER PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
FIG. 8. LDA Band structure of Lax Sr 1−x TiO3 for different values of x. DFT and MLWF bands are shown in gray and in color (from blue
to red), respectively. While the general shape and bandwidth of the Ti-t2g remain basically constant for the whole series, its relative position
with respect to EF moves gradually to lower energies for increasing x, marking the expected filling of the t2g bands from d 0 for SrTiO3 (x = 0)
to d 1 in LaTiO3 (x = 1).
the occupation of the vacancy is decreased with increasing to ω = 0 yield the electron occupation of the corresponding
Ti-OV -Ti distance, while the Ti occupancies increase pro- orbitals, and in our simulations all have values consistent
portionally. The most pronounced change in occupation is with a homogeneous distribution of the excess charge of one
observed for U (OV ) =U (Ti) (Fig. 7, lower panel). In this electron per La ion into the Ti bands. Increasing the strength
case, the vacancy site occupation drops from close to 2 (for the of the interaction parameter from U = 4 to 8 eV does not
shortest considered Ti-OV -Ti distance of 0.92) down to 1 (for have a noticeable effect on A(ω). However, the correspond-
distances equal to or larger than 1.04). This case corresponds ing quasiparticle weights, Z [Fig. 9(b)], are reduced from
to the site-selective Mott-insulating state obtained previously the “uncorrelated” value of Z = 1, indicating some degree
in Sec. III C, in which the vacancy occupation is essentially of electronic correlation in the corresponding bands. Z de-
locked to 1 due to the gap in the corresponding local spectral creases, and becomes more sensitive to the value of U , as the
function. The strong dependence of the DMFT occupancies concentration of La increases, consistent with the proximity
on U (OV ) for larger Ti-OV -Ti distances is of course consistent to the Mott-insulating state at x = 1. Thus, our results confirm
with the results already presented in Fig. 3, where the structure the validity of the rigid-band picture under LaSr substitution,
was relaxed within LDA, leading to a Ti-OV -Ti distance of while at the same time indicating moderate correlation effects,
1.024 in these units. increasing with La concentration.
These results indicate that the site-selective Mott-
insulating state, corresponding to a singly occupied vacancy IV. SUMMARY AND CONCLUSIONS
site, is favorable for an elongated Ti-OV -Ti distance. We note
that the lack of charge self-consistency does not allow us to We have presented a DFT+DMFT description of the long-
extract reliable DFT+DMFT total energies as a function of discussed problem of oxygen vacancies in SrTiO3 , focusing
the Ti-OV -Ti distance. on a controlled and systematic treatment of electronic correla-
tions on the vacancy site. This is achieved by using a minimal
correlated subspace, which consists of the low-energy Ti-t2g
F. Comparison with LaSr substitution orbitals plus an additional Wannier function located on the
As we mentioned in the Introduction, the question of how vacancy site. Our study reveals a strong influence of U (OV )
different doping sources affect the detailed electronic proper- on the overall electronic structure of the system. Furthermore,
ties is an important one, with earlier studies suggesting that,
while OV doping significantly alters the band structure as we
have seen here, LaSr causes only a rigid shift of the bands
[20,76]. Here, we test whether the DFT+DMFT treatment
is consistent with these earlier findings by calculating the
behavior when a small concentration of Sr ions is replaced
by La.
Figure 8 shows the evolution of the DFT Lax Sr 1−x TiO3
band structure for three different values of x corresponding to
La concentrations of less than or equal to half. These results
confirm that the main features of the SrTiO3 band structure
are robust against doping. The only, yet rather important,
difference between the three cases is the increasing partial
filling of the Ti-t2g bands with increasing La concentration. FIG. 9. (a) Ti-t2g DMFT spectral functions and (b) correspond-
The DFT+DMFT spectral functions of the Ti-t2g bands for ing quasiparticle weights for three different concentrations x of
these three La concentrations, shown in Fig. 9(a), confirm Lax Sr 1−x TiO3 for a 40-atom unit cell and two different values of U .
the metallicity of the system. Their respective integrals up Spectral functions in (a) are shifted on the y-axis for clarity.
023027-8OXYGEN VACANCIES IN STRONTIUM TITANATE: A … PHYSICAL REVIEW RESEARCH 3, 023027 (2021)
our cRPA calculations support the importance of electronic Lastly, we have compared OV -doped SrTiO3 with the case
correlations on the OV site, yielding a U (OV ) close to 60% of LaSr substitution, which represents another common way of
of U (Ti). electron doping the system. Our DFT+DMFT results, in line
In particular, we find a transition from a doubly occu- with previous DFT findings, show that Lax Sr 1−x TiO3 is much
pied [U (OV ) = 0] to a singly occupied OV state [U (OV ) = closer to the trivial doping case, with no significant change
U (Ti)], accompanied by a charge transfer of one electron from in the band structure other than an increasing uniform filling
the defect state into the conduction band. The latter state is of the Ti-t2g bands by the extra electrons donated by the La
equivalent to the scenario proposed by Lin and Demkov based cations. This demonstrates that different routes to electron-
on a minimal model of the vacancy levels [38], and con- doping in SrTiO3 are certainly not equivalent, and can lead to
sistent with the spin-polarized GGA+U results of Hou and rather different electronic structures.
Terakura [31], but without the need for an artificial spin- Finally, we mention that the method that we have demon-
symmetry breaking. It can in principle reconcile apparently strated for controlling and analyzing the strength of electronic
contradictory experimental observations of low-temperature correlation on a vacancy defect state within the DFT+DMFT
metallic conductivity in combination with localized in-gap formalism is applicable to other defects beyond the oxygen
states [24,25,28,29]. Our results are also consistent with a vacancies studied here. Our finding that the electronic na-
recent DFT+DMFT study of oxygen vacancies at the SrTiO3 ture of the vacancy state depends strongly on the strength of
surface [39], which also found metallicity in combination with the local interaction, U (OV ), highlights the need for such a
a split-off spectral feature inside the gap. methodology. We hope that, in addition to contributing to the
Analysis of the effect of the distance between two ongoing debate about the case of doped SrTiO3 , our work will
Ti ions separated by an OV on the nature of the OV charge motivate similar studies on other materials in which correlated
leads to two important conclusions. First, different treatments defects might play an important role.
of electronic interactions at the DFT(+U ) level lead to dif-
ferent Ti-OV -Ti distances. This in turn affects the relative
OV and Ti-t2g energy levels, with smaller Ti-OV -Ti distances
ACKNOWLEDGMENTS
favoring localization of the charge on the OV and larger dis-
tances promoting partial filling of the Ti orbitals with one of We thank Peitao Liu and Cesare Franchini for fruitful
the electrons released by the missing oxygen. Second, this discussions and technical help with the cRPA calculations.
picture is largely maintained after a DFT+DMFT analysis, This work was supported by the Swiss National Science
with the OV electron occupation also depending on U (OV ). Foundation through NCCR-MARVEL and by the Körber
Future DFT+DMFT calculations considering full charge self- Foundation. Calculations have been performed on the cluster
consistency and structural relaxations could provide further “Piz Daint,” hosted by the Swiss National Supercomputing
insight as to which Ti-OV -Ti distance and vacancy site occu- Centre and supported under project IDs s889 (User Lab) and
pation is indeed energetically preferred. mr26 (MARVEL), and the “Euler” cluster of ETH Zurich.
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